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G = GL2(𝔽3)⋊D5order 480 = 25·3·5

1st semidirect product of GL2(𝔽3) and D5 acting via D5/C5=C2

non-abelian, soluble

Aliases: Dic5.2S4, GL2(𝔽3)⋊1D5, SL2(𝔽3)⋊1D10, C2.7(D5×S4), C10.4(C2×S4), Q8⋊D154C2, Q8.4(S3×D5), (C5×Q8).4D6, C51(C4.3S4), Q82D53S3, Dic5.A43C2, (C5×GL2(𝔽3))⋊1C2, (C5×SL2(𝔽3))⋊1C22, SmallGroup(480,970)

Series: Derived Chief Lower central Upper central

C1C2Q8C5×SL2(𝔽3) — GL2(𝔽3)⋊D5
C1C2Q8C5×Q8C5×SL2(𝔽3)Dic5.A4 — GL2(𝔽3)⋊D5
C5×SL2(𝔽3) — GL2(𝔽3)⋊D5
C1C2

Generators and relations for GL2(𝔽3)⋊D5
 G = < a,b,c,d,e,f | a4=c3=d2=e5=f2=1, b2=a2, bab-1=faf=dbd=a-1, cac-1=ab, dad=fbf=a2b, ae=ea, cbc-1=a, be=eb, dcd=c-1, ce=ec, fcf=ac, de=ed, df=fd, fef=e-1 >

Subgroups: 818 in 84 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22 [×5], C5, S3 [×2], C6, C8 [×2], C2×C4, D4 [×4], Q8, C23, D5 [×2], C10, C10, C12, D6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, D10 [×4], C2×C10, SL2(𝔽3), D12, C5×S3, D15, C30, C8⋊C22, C52C8, C40, C4×D5, D20 [×2], C5⋊D4, C5×D4, C5×Q8, C22×D5, GL2(𝔽3), GL2(𝔽3), C4.A4, C3×Dic5, S3×C10, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, C4.3S4, C5⋊D12, C5×SL2(𝔽3), D40⋊C2, C5×GL2(𝔽3), Q8⋊D15, Dic5.A4, GL2(𝔽3)⋊D5
Quotients: C1, C2 [×3], C22, S3, D5, D6, D10, S4, C2×S4, S3×D5, C4.3S4, D5×S4, GL2(𝔽3)⋊D5

Character table of GL2(𝔽3)⋊D5

 class 12A2B2C2D34A4B5A5B68A8B10A10B10C10D12A12B15A15B20A20B30A30B40A40B40C40D
 size 1112306086102281260222424404016161212161612121212
ρ111111111111111111111111111111    trivial
ρ211-1-1111-1111-1111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ3111-1-111-11111-11111-1-11111111111    linear of order 2
ρ411-11-1111111-1-111-1-111111111-1-1-1-1    linear of order 2
ρ5220-20-12-222-100220011-1-122-1-10000    orthogonal lifted from D6
ρ622020-12222-1002200-1-1-1-122-1-10000    orthogonal lifted from S3
ρ722-200220-1-5/2-1+5/22-20-1+5/2-1-5/21-5/21+5/200-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ822200220-1+5/2-1-5/2220-1-5/2-1+5/2-1-5/2-1+5/200-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ922-200220-1+5/2-1-5/22-20-1-5/2-1+5/21+5/21-5/200-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ1022200220-1-5/2-1+5/2220-1+5/2-1-5/2-1+5/2-1-5/200-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ113311-10-1-3330-1133110000-1-100-1-1-1-1    orthogonal lifted from C2×S4
ρ12331-110-13330-1-133110000-1-100-1-1-1-1    orthogonal lifted from S4
ρ1333-1-1-10-133301133-1-10000-1-1001111    orthogonal lifted from S4
ρ1433-1110-1-33301-133-1-10000-1-1001111    orthogonal lifted from C2×S4
ρ154-4000-20044200-4-40000-2-200220000    orthogonal lifted from C4.3S4
ρ1644000-240-1-5-1+5-200-1+5-1-500001+5/21-5/2-1-5-1+51-5/21+5/20000    orthogonal lifted from S3×D5
ρ1744000-240-1+5-1-5-200-1-5-1+500001-5/21+5/2-1+5-1-51+5/21-5/20000    orthogonal lifted from S3×D5
ρ184-400010044-100-4-4003-31100-1-10000    orthogonal lifted from C4.3S4
ρ194-400010044-100-4-400-331100-1-10000    orthogonal lifted from C4.3S4
ρ204-4000-200-1-5-1+52001-51+500001+5/21-5/200-1+5/2-1-5/2ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ5    orthogonal faithful
ρ214-4000-200-1-5-1+52001-51+500001+5/21-5/200-1+5/2-1-5/283ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5    orthogonal faithful
ρ224-4000-200-1+5-1-52001+51-500001-5/21+5/200-1-5/2-1+5/2ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52    orthogonal faithful
ρ234-4000-200-1+5-1-52001+51-500001-5/21+5/200-1-5/2-1+5/2ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ583ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52    orthogonal faithful
ρ24662000-20-3-35/2-3+35/20-20-3+35/2-3-35/2-1+5/2-1-5/200001+5/21-5/2001+5/21+5/21-5/21-5/2    orthogonal lifted from D5×S4
ρ2566-2000-20-3+35/2-3-35/2020-3-35/2-3+35/21+5/21-5/200001-5/21+5/200-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5×S4
ρ26662000-20-3+35/2-3-35/20-20-3-35/2-3+35/2-1-5/2-1+5/200001-5/21+5/2001-5/21-5/21+5/21+5/2    orthogonal lifted from D5×S4
ρ2766-2000-20-3-35/2-3+35/2020-3+35/2-3-35/21-5/21+5/200001+5/21-5/200-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5×S4
ρ288-8000200-2+25-2-25-2002+252-250000-1+5/2-1-5/2001+5/21-5/20000    orthogonal faithful, Schur index 2
ρ298-8000200-2-25-2+25-2002-252+250000-1-5/2-1+5/2001-5/21+5/20000    orthogonal faithful, Schur index 2

Smallest permutation representation of GL2(𝔽3)⋊D5
On 80 points
Generators in S80
(1 54 12 69)(2 55 13 70)(3 51 14 66)(4 52 15 67)(5 53 11 68)(6 19 60 64)(7 20 56 65)(8 16 57 61)(9 17 58 62)(10 18 59 63)(21 34 78 47)(22 35 79 48)(23 31 80 49)(24 32 76 50)(25 33 77 46)(26 45 74 40)(27 41 75 36)(28 42 71 37)(29 43 72 38)(30 44 73 39)
(1 42 12 37)(2 43 13 38)(3 44 14 39)(4 45 15 40)(5 41 11 36)(6 21 60 78)(7 22 56 79)(8 23 57 80)(9 24 58 76)(10 25 59 77)(16 49 61 31)(17 50 62 32)(18 46 63 33)(19 47 64 34)(20 48 65 35)(26 67 74 52)(27 68 75 53)(28 69 71 54)(29 70 72 55)(30 66 73 51)
(16 23 49)(17 24 50)(18 25 46)(19 21 47)(20 22 48)(26 52 45)(27 53 41)(28 54 42)(29 55 43)(30 51 44)(31 61 80)(32 62 76)(33 63 77)(34 64 78)(35 65 79)(36 75 68)(37 71 69)(38 72 70)(39 73 66)(40 74 67)
(6 60)(7 56)(8 57)(9 58)(10 59)(16 23)(17 24)(18 25)(19 21)(20 22)(26 74)(27 75)(28 71)(29 72)(30 73)(36 53)(37 54)(38 55)(39 51)(40 52)(41 68)(42 69)(43 70)(44 66)(45 67)(61 80)(62 76)(63 77)(64 78)(65 79)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)
(1 46)(2 50)(3 49)(4 48)(5 47)(6 75)(7 74)(8 73)(9 72)(10 71)(11 34)(12 33)(13 32)(14 31)(15 35)(16 44)(17 43)(18 42)(19 41)(20 45)(21 68)(22 67)(23 66)(24 70)(25 69)(26 56)(27 60)(28 59)(29 58)(30 57)(36 64)(37 63)(38 62)(39 61)(40 65)(51 80)(52 79)(53 78)(54 77)(55 76)

G:=sub<Sym(80)| (1,54,12,69)(2,55,13,70)(3,51,14,66)(4,52,15,67)(5,53,11,68)(6,19,60,64)(7,20,56,65)(8,16,57,61)(9,17,58,62)(10,18,59,63)(21,34,78,47)(22,35,79,48)(23,31,80,49)(24,32,76,50)(25,33,77,46)(26,45,74,40)(27,41,75,36)(28,42,71,37)(29,43,72,38)(30,44,73,39), (1,42,12,37)(2,43,13,38)(3,44,14,39)(4,45,15,40)(5,41,11,36)(6,21,60,78)(7,22,56,79)(8,23,57,80)(9,24,58,76)(10,25,59,77)(16,49,61,31)(17,50,62,32)(18,46,63,33)(19,47,64,34)(20,48,65,35)(26,67,74,52)(27,68,75,53)(28,69,71,54)(29,70,72,55)(30,66,73,51), (16,23,49)(17,24,50)(18,25,46)(19,21,47)(20,22,48)(26,52,45)(27,53,41)(28,54,42)(29,55,43)(30,51,44)(31,61,80)(32,62,76)(33,63,77)(34,64,78)(35,65,79)(36,75,68)(37,71,69)(38,72,70)(39,73,66)(40,74,67), (6,60)(7,56)(8,57)(9,58)(10,59)(16,23)(17,24)(18,25)(19,21)(20,22)(26,74)(27,75)(28,71)(29,72)(30,73)(36,53)(37,54)(38,55)(39,51)(40,52)(41,68)(42,69)(43,70)(44,66)(45,67)(61,80)(62,76)(63,77)(64,78)(65,79), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,46)(2,50)(3,49)(4,48)(5,47)(6,75)(7,74)(8,73)(9,72)(10,71)(11,34)(12,33)(13,32)(14,31)(15,35)(16,44)(17,43)(18,42)(19,41)(20,45)(21,68)(22,67)(23,66)(24,70)(25,69)(26,56)(27,60)(28,59)(29,58)(30,57)(36,64)(37,63)(38,62)(39,61)(40,65)(51,80)(52,79)(53,78)(54,77)(55,76)>;

G:=Group( (1,54,12,69)(2,55,13,70)(3,51,14,66)(4,52,15,67)(5,53,11,68)(6,19,60,64)(7,20,56,65)(8,16,57,61)(9,17,58,62)(10,18,59,63)(21,34,78,47)(22,35,79,48)(23,31,80,49)(24,32,76,50)(25,33,77,46)(26,45,74,40)(27,41,75,36)(28,42,71,37)(29,43,72,38)(30,44,73,39), (1,42,12,37)(2,43,13,38)(3,44,14,39)(4,45,15,40)(5,41,11,36)(6,21,60,78)(7,22,56,79)(8,23,57,80)(9,24,58,76)(10,25,59,77)(16,49,61,31)(17,50,62,32)(18,46,63,33)(19,47,64,34)(20,48,65,35)(26,67,74,52)(27,68,75,53)(28,69,71,54)(29,70,72,55)(30,66,73,51), (16,23,49)(17,24,50)(18,25,46)(19,21,47)(20,22,48)(26,52,45)(27,53,41)(28,54,42)(29,55,43)(30,51,44)(31,61,80)(32,62,76)(33,63,77)(34,64,78)(35,65,79)(36,75,68)(37,71,69)(38,72,70)(39,73,66)(40,74,67), (6,60)(7,56)(8,57)(9,58)(10,59)(16,23)(17,24)(18,25)(19,21)(20,22)(26,74)(27,75)(28,71)(29,72)(30,73)(36,53)(37,54)(38,55)(39,51)(40,52)(41,68)(42,69)(43,70)(44,66)(45,67)(61,80)(62,76)(63,77)(64,78)(65,79), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,46)(2,50)(3,49)(4,48)(5,47)(6,75)(7,74)(8,73)(9,72)(10,71)(11,34)(12,33)(13,32)(14,31)(15,35)(16,44)(17,43)(18,42)(19,41)(20,45)(21,68)(22,67)(23,66)(24,70)(25,69)(26,56)(27,60)(28,59)(29,58)(30,57)(36,64)(37,63)(38,62)(39,61)(40,65)(51,80)(52,79)(53,78)(54,77)(55,76) );

G=PermutationGroup([(1,54,12,69),(2,55,13,70),(3,51,14,66),(4,52,15,67),(5,53,11,68),(6,19,60,64),(7,20,56,65),(8,16,57,61),(9,17,58,62),(10,18,59,63),(21,34,78,47),(22,35,79,48),(23,31,80,49),(24,32,76,50),(25,33,77,46),(26,45,74,40),(27,41,75,36),(28,42,71,37),(29,43,72,38),(30,44,73,39)], [(1,42,12,37),(2,43,13,38),(3,44,14,39),(4,45,15,40),(5,41,11,36),(6,21,60,78),(7,22,56,79),(8,23,57,80),(9,24,58,76),(10,25,59,77),(16,49,61,31),(17,50,62,32),(18,46,63,33),(19,47,64,34),(20,48,65,35),(26,67,74,52),(27,68,75,53),(28,69,71,54),(29,70,72,55),(30,66,73,51)], [(16,23,49),(17,24,50),(18,25,46),(19,21,47),(20,22,48),(26,52,45),(27,53,41),(28,54,42),(29,55,43),(30,51,44),(31,61,80),(32,62,76),(33,63,77),(34,64,78),(35,65,79),(36,75,68),(37,71,69),(38,72,70),(39,73,66),(40,74,67)], [(6,60),(7,56),(8,57),(9,58),(10,59),(16,23),(17,24),(18,25),(19,21),(20,22),(26,74),(27,75),(28,71),(29,72),(30,73),(36,53),(37,54),(38,55),(39,51),(40,52),(41,68),(42,69),(43,70),(44,66),(45,67),(61,80),(62,76),(63,77),(64,78),(65,79)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80)], [(1,46),(2,50),(3,49),(4,48),(5,47),(6,75),(7,74),(8,73),(9,72),(10,71),(11,34),(12,33),(13,32),(14,31),(15,35),(16,44),(17,43),(18,42),(19,41),(20,45),(21,68),(22,67),(23,66),(24,70),(25,69),(26,56),(27,60),(28,59),(29,58),(30,57),(36,64),(37,63),(38,62),(39,61),(40,65),(51,80),(52,79),(53,78),(54,77),(55,76)])

Matrix representation of GL2(𝔽3)⋊D5 in GL6(𝔽241)

100000
010000
000010
000001
00240000
00024000
,
100000
010000
00024000
001000
000001
00002400
,
100000
010000
001000
00002400
00000240
000100
,
24000000
02400000
001000
000010
000100
00000240
,
010000
2401890000
001000
000100
000010
000001
,
161730000
642250000
009999990
0099142099
00990142142
0009914299

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,240,0,0,0,0,0,0,240,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,240],[0,240,0,0,0,0,1,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,64,0,0,0,0,173,225,0,0,0,0,0,0,99,99,99,0,0,0,99,142,0,99,0,0,99,0,142,142,0,0,0,99,142,99] >;

GL2(𝔽3)⋊D5 in GAP, Magma, Sage, TeX

{\rm GL}_2({\mathbb F}_3)\rtimes D_5
% in TeX

G:=Group("GL(2,3):D5");
// GroupNames label

G:=SmallGroup(480,970);
// by ID

G=gap.SmallGroup(480,970);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,3389,93,1347,2111,3168,172,1272,1909,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=c^3=d^2=e^5=f^2=1,b^2=a^2,b*a*b^-1=f*a*f=d*b*d=a^-1,c*a*c^-1=a*b,d*a*d=f*b*f=a^2*b,a*e=e*a,c*b*c^-1=a,b*e=e*b,d*c*d=c^-1,c*e=e*c,f*c*f=a*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

Export

Character table of GL2(𝔽3)⋊D5 in TeX

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